by standard estimates for solutions to BSDEPs (see [1]}, [2]}, [3]} and its derivative under our assumptions, which can be regarded as a special case of Theorem 2.3 in the following), we have
\(\begin{aligned}&E\bigg [\sup _{0\le t\le T}|\hat{y}_t|^2+\int _0^T|\hat{z}_t|^2dt+\int _0^T\int _{\mathcal {E}}|\hat{\tilde{z}}_{(t,e)}|^2\nu (de)dt\bigg ]\\&\le C E\bigg [\big |\phi (x_{1,T})-\phi (x_{2,T})\big |^2+\int _0^T\big |g(t,x_1,y_1,z_1,\tilde{z}_1)-g(t,x_2,y_1,z_1,\tilde{z}_1)\big |^2dt\bigg ]\\&\le C E\bigg [|C_\phi \hat{x}_T|^2+\int _0^TC^2_g|\hat{x}|^2dt\bigg ]\le C(C^2_\phi +C^2_gT)E\bigg [\sup _{0\le t\le T}|\hat{x}_t|^2\bigg ].\end{aligned}\) 
